You take a standard deck of 52 cards (four suits: spades, hearts, clubs, diamonds; thirteen ranks: A, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K). The deck is shuffled uniformly. You go through the shuffled deck one card at a time and count the number of times that two consecutive cards have the same rank. For instance, if you see A, A, 5, K, Q, Q, Q, then you see two consecutive aces, then two consecutive queens (the first two queens), and lastly another pair of two consecutive queens (the last two queens).

(a) For k = 1, ,...,51, let X_k be a random variable that equals 1 if the kth and (k+1)th cards in the deck have the same rank and zero otherwise. What is the expectation of X_k?

(b) Let X be the total number of times that two consecutive cards have the same rank. Find the expectation of X.